thank Professors Chih-Ming Ho and Yu-Chong Tai for suggesting this problem and, along with Dr. Joon Mo. Yang, discussing their results with us. 484. 10-1.
Microfilter Simulations and Scaling Laws David R. Mott, Elaine S. Oran, and Carolyn R. Kaplan Laboratory for Computational Physics and Fluid Dynamics U. 5. Naval Research Laboratory, Washington, B.C. 20375
Abstract. This work presents DSMC simulations used to quantify the effect of Knudsen number on flows through filters with micron-scale holes. The empirical scaling laws currently used to predict pressure drop as a function of flow conditions and filter geometry are based on experiments and calculations within the continuum regime. We illustrate that noncontinuum effects are significant for filters designed to capture biowarfare agents and pathogens of current interest. We also suggest a scaling based on Knudsen number for correcting the classic scaling laws to include these effects.
I
INTRODUCTION
Biological agents and other airborne contaminants may be isolated by a filtering process in which air is pulled through grates with micron-sized holes. Filters with holes that are smaller than the target particles are used so that these particles do not pass through the filter and can be gathered at the filter surface. Such filtering processes require that large amounts of air be filtered since the concentrations of these agents can be minute. Unfortunately, filtering such large volumes of air becomes costly since the power requirement for filtering is the product of the volume flow rate and the pressure drop across the filter that induces this flowrate. Therefore, filters must be designed to minimize this pressure drop to enhance efficiency. Conventional filters appear in a variety of practical applications, from isolating particulates in a gas sample to controlling turbulence levels in a wind tunnel. Researchers have long tried to predict filter performance based on some global geometric properties and flowfield conditions [1,2]. These studies led to "scaling laws," or algebraic expressions which indicate how the pressure drop across a screen or filter scales with these parameters. Most recently, experimental and computational studies have been conducted in the continuum regime for filters with circular holes with diameters from 5 to 12 microns [3,4], which is well outside the parameter range of previous work. These studies yielded empirical relationships between pressure drop and flow rate in terms of two geometric parameters and one flow parameter: the ratio of hole area to filter area (which is called the opening factor, /?), the ratio of filter thickness to hole diameter, t/d, and the Reynolds number, Re, given by
The density, velocity, and viscosity of the inflow are given by /?2-n, [/,-„ and /^-n, respectively. This definition of Re includes the blockage effect of the filter by scaling [/,-„ by /?. Using the effective flow through the filter holes in the definition for Re improves the correlation of results for various hole geometries and flow conditions [2]. In these recent studies, initial computational results did not agree well with experiment [3]. However, closer inspection of the filters revealed discrepancies between the intended hole geometry (which was used in the computations) and that which was actually fabricated. Calculations performed using a more accurate hole geometry agreed well with experiment, and led to the expression [4] (10/Ee + 0.22)
(2)
for the nondimensional pressure drop, K, across the filter. Equation (2) indicates that K is inversely proportional to the square of the opening factor, linear in t/d, and linear in I/Re. CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis 2001 American Institute of Physics 0-7354-0025-3 480
Equation (2) was developed using solutions to the Navier-Stokes equations and then compared to experimental results. The calculations included noncontinuum effects in the form of a slip boundary condition at the filter surface, but these effects were minor for the cases tested. Rather than include an additional parameter such as the Knudsen number, Kn, in the scaling law, these effects have been incorporated into the values of the constants in Eq. (2). Filters for biodetection, however, will require holes that are even smaller than those tested in these studies. As hole diameters are reduced to 1 micron or less, Knudsen-number effects will be substantial, and a continuum Navier-Stokes solver will not provide accurate predictions. Furthermore, these effects must be included explicitly in the scaling law. This paper describes current two-dimensional DSMC calculations used to quantify the relationship between power consumption and flow rate for filters in the high Knudsen number regime. These simulations led to a refinement in the scaling laws based on Knudsen number which expands the range in which these expressions accurately predict filter performance.
II
NUMERICAL APPROACH
Two-dimensional DSMC calculations were first performed for a "baseline" case described in Fig. 1, and then parameter studies were performed by varying the flow conditions are geometry relative to this case. The
Symmetry Boundary
1.5 Jim
-7 t = 3 jim
/ Diffuse Wall at Local Temperature
Symmetry Boundary FIGURE 1. Baseline geometry.
baseline case consisted of a filter three microns thick with one micron holes spaced four microns apart. These dimensions give t/d = 3 and /3 = 1/4. For the current studies, we fix /3 and t/d so that variation in the results is limited to Re effects and Kn effects. The domain shown in Fig. 1 exploits planes of symmetry along the centerline of a filter hole and along the boundary between adjacent holes to reduce the computational expense of each calculation. The baseline inflow conditions are air at roughly sea-level temperature and density, flowing at 4 m/s towards the filter from the left. The Knudsen number for this baseline case, defined as
Kn = ~~ characteristic length
d
in terms of the mean free path of the inflow, A« n , is 0.0576. The code used for these calculations is a modified version of Bird's DSMC2.FOR [5]. The modified code allows the user to include an arbitrary number of surfaces within the domain and impose an adiabatic, diffusereflection condition at the surface. The filter is then modeled as a step-shaped blockage as shown in Fig. 1. In order to control Re from case to case, the mass flux through the domain was specified by the inflow conditions. The pressure drop across the filter is then measured at the end of each simulation. Particles entering from the downstream side of the domain were sampled from a prescribed distribution. This condition overspecifies the subsonic outflow, and close to the exit plane the solution adjusts in an attempt to resolve the inconsistency. By placing the downstream boundary sufficiently far away from the filter, we prevent this effect from corrupting the calculation near the filter.
Ill
RESULTS AND DISCUSSION
Contour plots of the x- velocity and the density for the baseline case are shown in Figs. 2 and 3. The velocity plot shows how the flow accelerates through the hole in order to maintain mass flux, reaching a maximum speed along the hole centerline around 24 m/s. The flow also exhibits a distinct boundary-layer structure as it passes
481
y (µm) y (µm)
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22
44
66
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00 -2-2 -10 -10
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FIGURE2.2.2.Contours Contoursofofofx-velocity x-velocityfor forthe thebaseline baselinecase case FIGURE FIGURE Contours x-velocity for the baseline case
y (µm) y (µm)
DENSITY: 1.06 1.061.07 1.071.08 1.081.09 1.091.10 1.101.11 1.111.12 1.121.13 1.131.14 1.141.15 1.151.16 1.161.17 1.17 1.18 DENSITY: DENSITY: 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.171.18 1.18 2 2
00 -2-2 -10 -10
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FIGURE3.3.3.Density Densitycontours contoursfor forthe thebaseline baselinecase. case. FIGURE FIGURE Density contours for the baseline case. throughthe thechannel. channel. The Thedensity densityplot plotindicates indicatesaaaslight slightincrease increaseinin inthe thedensity densityasas asthe the lter approached. through through the channel. The density plot indicates slight increase the density the lter filterisis isapproached. approached. This\ram \rame ect" e ect"isisisproduced producedby byour ourforcing forcingthe the ow
owinininatatataaaspeci ed speci edmass mass ux.
ux.This Thisincrease increaseofof ofaround around 2% This This "ram effect" produced by our forcing the flow specified mass flux. This increase around2% 2% in the density is followed by an expansion through the lter that gives a nal density value approximately 10% inin the the density density isis followed followed by by an an expansion expansionthrough throughthe the lter filter that thatgives givesaa nal finaldensity densityvalue valueapproximately approximately10% 10% belowthat thatofofofthe theincoming incoming ow.
ow. below below that the incoming flow. Plots of the pressure, x-velocity, andthe thedensity densityare aregiven giveninininFig. Fig.4.4.4.The Thecurves curvesfor forthe thedensity densityand andx-velocity x-velocity Plots of the pressure, x-velocity, Plots of the pressure, x-velocity,and and the density are given Fig. The curves for the density and x-velocity mirror the contour plots above. The noise seen in these curves is due to the disparity between the thermal mirror the contour plots above. The noise seen in these curves is due to the disparity between the mirror the contour plots above. The noise seen in these curves is due to the disparity between thethermal thermal velocity of the particles in the simulation, which is on the order of 500 m/s, and the mean
ow velocity. Even velocity of the particles in the simulation, which is on the order of 500 m/s, and the mean
ow velocity. velocity of the particles in the simulation, which is on the order of 500 m/s, and the mean flow velocity.Even Even with this noise, however, the plateaus in pressure in front of and behind the lter are easy to identify, and the with this noise, however, the plateaus in pressure in front of and behind the lter are easy to identify, and with this noise, however, the plateaus in pressure in front of and behind the filter are easy to identify, andthe the di erence in these levels is the value of p that our scaling laws attempt to predict. di erence p that difference in in these these levels levels isis the the value value of ofAp that our ourscaling scalinglaws lawsattempt attempttotopredict. predict. Theresults resultsofofofthe thebaseline baselinecase case and andseveral several other othercases cases obtained obtainedbyby byvarying varying the the ow
ow conditions and the scale The The results the baseline caseand severalother casesobtained varyingthe flowconditions conditionsand andthe thescale scale the geometry are included in Fig. 5. Figure 5 also includes Yang et al.'s scaling law from Eq. (2), as well as ofofofthe geometry are included in Fig. 5. Figure 5 also includes Yang et al.'s scaling law from Eq. (2), as well the geometry are included in Fig. 5. Figure 5 also includes Yang et al.'s scaling law from Eq. (2), as wellasas some of their experimental data [4]. The experimental data is for cases in the range 0 :0057Kn Kn0:00127. :0127.InIn some of their experimental data [4]. The experimental data is for cases in the range 0 : 0057 some of their experimental data [4]. The experimental data is for cases in the range 0.0057 < Kn < 0.0127. In Fig5,5,squares squaresindicate indicatedata datafor forconstant constant Kn Kn==0:00576 :0576 as de nedininEq. Eq. (3). (3).These These points pointslielie below the curve Fig Fig 5, squares indicate data for constant Kn = 0.0576asasde ned defined in Eq.(3). Thesepoints liebelow belowthe thecurve curve for Eq. (2) by around a factor of 2, but the experimental data lie below the curve as well at low Re. .Considering Considering for Eq. (2) by around a factor of 2, but the experimental data lie below the curve as well at low Re for Eq. (2) by around a factor of 2, but the experimental data lie below the curve as well at low Re. Considering thatthe thecurrent currentsimulations simulationsare aretwo-dimensional two-dimensionalcompared compared to the thethree-dimensional three-dimensional data dataofof ofYang Yang et al., and that that the current simulations are two-dimensional comparedtotothe three-dimensionaldata Yangetetal., al.,and and that we are operating well outside the parameter range in which the empirical t was derived, agreement is that that we we are are operating operating well well outside outside the the parameter parameter range rangeininwhich whichthe the empirical empirical tfitwas wasderived, derived,agreement agreementisis good. A linear dependence of Konon1=Re 1 =Reisisseen seen in the Kn==0:00576 :0576data, data, which reinforces the form used in good. A linear dependence of K in the Kn which reinforces the form used in good. A linear dependence of K on I/Re is seen in the Kn = 0.0576 data, which reinforces the form used in Eq.(2) (2)for forthe theRe Re term. Eq. Eq. (2) for the Re term. term. The circular data pointsinininFig. Fig. 55correspond correspondtototoKn Kn =0.0309, 0.0309,0.117, 0.117, and and0.233 0.233asas asone one travels travelsaway away form The circular data points form The circular data points Fig. 5indicate correspond Kn == 0.117,and 0.233 onetravels away form the empirical curve. These points that varying Kn0.0309, while holding the parameters in Eq. (2) constant the empirical curve. These points indicate that varying Kn while holding the parameters in Eq. (2) constant the empirical curve. These points indicate that varying Kn while holding the parameters in Eq. (2) constant resultsinininsigni cant signi cantdi erences di erencesinininKKKthat thatthe thescaling scalinglaw lawcannot cannotpredict. predict. Consider Considerthe thecircular circular data point results point results significant differences that the scaling law cannot predict. Consider the circulardata data point farthest from the empirical curve, for which Kn = 0 : 233 and Re = 0 : 68. If K showed no dependence on Kn, , farthest from the empirical curve, for which Kn == 00.233 :233 and Re ==00.68. :68. IfIfKK showed nonodependence ononKn farthest from the empirical curve, for which Kn and Re showed dependence Kn, then this Kn = 0 : 233 data point would fall in line with the Kn = 0 : 0576 data. This is clearly not the case; the then this Kn == 00.233 :233 data point would fall ininline with the Kn ==00.0576 :0576 data. This isisclearly not the case; the then this Kn data point would fall line with the Kn data. This clearly not the case; the valueofofKKinterpolated interpolatedatatReRe==0:068:68from fromthe theKn Kn==0:00576 :0576data dataisisapproximately approximatelythree threetimes timesthat thatgiven givenbybythe the value value of K interpolated at Re = 0.68 from the Kn = 0.0576 data is approximately three times that given by the simulationforforKn Kn==0:0233 :233atatthe thesame sameReRe. .The Thesimulation simulationresults resultsindicate indicatethat thatasasKn Knincreases, increases,KKdecreases. decreases. simulation simulation for Kn = 0.233 at the same Re. The simulation results indicate that as Kn increases, K decreases. Therefore,ififwewewere weretotorun runmore moresimulations simulationsforforKn Kn!!0,0,the theresulting resultingvalues valuesofofKKwill willlielieabove abovethe thecurrent current Therefore, Therefore, if we were toapproaching run more simulations for curve Kn —>•or0,possibly the resulting values of K will lie above the current datapoints, points, perhaps theempirical empirical providing aconsistent consistent continuation the data perhaps approaching the curve or possibly providing a continuation ofofthe data points, perhaps approaching theregion. empirical curve or possibly providing a consistent continuation of the experimental data into the lower Re experimental data into the lower Re region. experimental data into the in lower Re Canweweinclude include afactor factor Eq.(2) (2)toregion. toaccount accountforforthis thisreduction reductionininKKasasKn Knincreases? increases?IfIfwewebase basethis thisfactor factor Can a in Eq. Can we include a factor in Eq. (2) to account for this reduction in K as Kn increases? If we base this factor Knand andinclude includeititasas f fononKn / on Kn and include it as
482
ρp (kg/m (kg/m3))
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FIGURE FIGURE4. 4. Centerline Centerline pressure, pressure, velocity, velocity, and and density density for for the the baseline baseline case. case. K = f ;
2 0-2
t 33.5 :5 + 3 d
(10 =Re + :22) ; (lO/fle + 00.22),
(4) (4)
@f df < 0: @Kn dKn
(5) (5)
wewemust mustrequire requiref/ to tosatisfy satisfy ff(Kn (Kn == 0)0)== 11,;
and and
InInother otherwords, words,the thescaling scalinglaw lawshould should return return the the continuum continuum limit limit as as Kn Kn ! —»•0,0,and andf/ decreases decreases as asKn Kn increases increases from fromKn Kn ==0.0.AAsimple simpleexpression expression that that satis es satisfies both both of ofthese these requirements requirements and and has has only only one one free free parameter parameter is is a f/ == Kn aa++Kn
(6) (6)
Thisform form has has obvious obviousproblems, problems, such such as as predicting predicting aa zero zero pressure pressure drop drop for for free-molecular free-molecular ((Kn —»•1 oo) flow. This Kn ! ) ow. However,we wedo donot not intend intend to to push push the the approximation approximation that that far. far. Our Our goal goal is is to to provide provide aa rough However, rough rule-of-thumb rule-of-thumb for lters filtersthat that can canisolate isolatecurrent current biological biological agents, agents, which which pins pins us us to to aa range range around around the the 11 micron for micron scale scale under under atmosphericconditions. conditions. Additional Additionaltheoretical theoreticalanalysis, analysis, physical physical reasoning, reasoning, and and comparisons comparisons with with experimental experimental atmospheric andcomputational computationalresults results are arerequired required to to develop develop aa form form of of f/ with with aa wider wider range range of of applicability, applicability, and and will will be be and the focus focus ofoffuture future research. research. the By choosing choosing one one data data point point at at each each of of the the four four Knudsen Knudsen numbers numbers tested, tested, the the constant constant in By in Eq. Eq. (6) (6) was was determined. Using UsingEq. Eq.(2) (2)and and the the DSMC DSMC result result for for each each case, case, aa target target value determined. value for for /f isis obtained obtained for for each each Kn. Kn. Then,the the constant constant ccin in Eq. Eq. (6) (6) isisdetermined determined by by minimizing minimizing the the error error in in matching matching these these four four ((Kn,f) data Then, Kn; f ) data points. This This procedure procedure gives gives cc == 00.05765. close-up of of the the DSMC DSMC results results are are shown shown in in Figure Figure 6, 6, along points. :05765. AA close-up along withEq. Eq.(2) (2)and and the the scaled scaled version version Eq. Eq. (4) (4) evaluated evaluated at at the the four four values values of of Kn Kn that that correspond correspond to to the the DSMC with DSMC data.The Thescaling scalingprovides providesreasonable reasonableagreement agreement considering considering that that we weare are trying trying to to match match data data at at four four di erent different data. Knudsennumbers numberswith withone onefree free parameter parameter and and aacrude crude form form for for f/.. The The scaled scaled curves curves at at the the highest highest and Knudsen and lowest lowest
483
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co
10 1
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DSMC, DSMC, Kn Kn== 0.0576 0.0576 DSMC, DSMC, 0.0309 0.0309
